NORTH COBB HIGH SCHOOL AB CALCULUS AP PREREQUISITE PACKET

Transcription

1 NORTH COBB HIGH SCHOOL AB CALCULUS AP PREREQUISITE PACKET The AP Course: AP Calculus AB is a college level course covering material traditionally taught in the first semester of college calculus. The course is taught in one semester. BC Calculus, corresponding to Calculus II at most universities, is taught during the second semester. Students are encouraged to take both semesters of AP Calculus. The Prerequisite Packet: Students need a strong foundation to be ready for the rigorous work required throughout the class. Completing the prerequisite packet should prepare you for the material to be taught in the course. This packet consists of material studied during primarily AC Math 3 but also some topics covered in AC Math 1 and Math 2. Students should anticipate working approximately 10 hours to complete it properly. This packet includes A "Toolkit of Functions" - you should be familiar with each of the graphs A formula and identities section. These are for your reference and do not need to be memorized at this time. A unit circle template with which to practice your unit circle A list of skills that you will need for AP Calculus. If you feel you are weak in any of the areas, let us know and we will be glad to help you. Calculus Prerequisite Problems. Please show all work. Answers are provided at the end of the packet. Calculators: Students will be using a graphing calculator throughout the course. A graphing calculator is required on the AP test. We will be using a TI-84 in class, but a TI NSpire or TI-89 calculator are also good choices.

2 Toolkit of Functions Students should know the basic shape of these functions and be able to graph their transformations without the assistance of a calculator. Constant f(x) =a Cubic f(x) =x 3 Identity f(x) = x Square Root f(x) =JX Absolute Value f(x) =I x I Greatest Integer f(x) =[ x ] Reciprocal 1 f(x) = x Exponential f(x) =a" Quadratic f(x) =x 2 Logarithmic f(x) = Inx

3 Trig Functions f(x) = sinx f(x) = cosx f(x) = tanx Polynomial Functions: A function P is called a polynomial if P(x) == anx n + an_1x n-1 + Where n is a nonnegative integer and the numbers 00, 01, az, 2 + a 2x + ~x + ao an are constants. Even degree Leading coefficient sign Odd degree Leading coefficient sign Positive Negative Positive Negative Number of roots equals the degree of the polynomial. Number of x intercepts is less than or equal to the degree. Number of "bends" is less than or equal to (degree - 1).

4 Formulas and Identities Trig Formulas: Arc Length of a circle: L = r 8 or d L = -.Zrtr d 2 Area of a sector of a circle:area = - r 2e or Area = -.nr Solving parts of a triangle: a b c Law of Sines: --=--=- sina sinb sinc 2 Law of Cosines: a = b 2 +c 2-2bc cosa Area of a Triangle: b 2 =a 2 +c 2-2ac cosb 2 c = a 2 + b 2-2ab cosc Area =..!.. be sina or Area =..!.. ac sinb or Area =..!.. ab sine Hero's formula: Area = ~s(s - a)(s- b)(s - c),where s = semi perimeter Ambiguous Case: 8 is acute 8 is obtuse or right Compute: alt = adj. sin 8 opp :-:; adj No triangle opp < alt No triangle opp > adj 1 triangle opp = alt 1 triangle (right) opp>adj 1 triangle alt<opp<adj 2 triangles Does a triangle exist? Yes - when (difference of 2 sides) < (third side) < (Sum of 2 sides)

5 Formulas and Identities. continued Trig Identities: Reciprocal Identities: csca=- seca=- cota=- sina cosa tana sina cosa Quotient Identities: tan A=- cota=- cosa sin A Pythagorean Identities: sin-a + cos-a = 1 tan-a + 1 = sec-a 1 + cot-a = esc-a Sum and Difference Identities: sin(a + B) = sina cosb+cosa sinb cos(a + B) = cosa cosb - sina sinb sin(a - B) = sina cosb - cosa sinb cos(a - B) = cosa cosb + sina sinb tan A + tanb - tan A - tanb tan (A + B) = tan (A - B) I-tanAtanB 1+ tana tan B Double Angle Identities: 2 tan A sin(2a) = 2sinA cosa tan(2a) = --.,.---- I-tan 2 A cos(2a) = cos-a - sin-a cos(2a) = 2cos 2A -1 cos(2a) = 1 - Zsin-A Half Angle Identities:. A JI-COSA A l-cosa sm-=± cos A = ±Jl+cosA tan-=± cosa Polar Formulas: x = rcos8 Geometric Formulas: y = rsin8 Area of a trapezoid: A = J:.. h (b l + b ) 2 Area of a triangle: A = J:.. bh 2 2 Area of an equilateral triangle: A = J3 S2 4 Area if a circle: A = nr' Circumference of a circle: C = 2rcr or C = dn tan-1 Y = 8 + 1t X < 0 x

6 Unit Circle - Degrees and Radians o o D Place degree measures in the circles. tano = Place radian measure in the squares. cot G= _ Place (cos OI sing) in parenthesis outside the square. cscf = _ Place tane outside the parenthesis. seco> _

7 SKILLS NEEDED FOR CALCULUS I. Algebra: *A. Exponents (operations with integer, fractional, and negative exponents) *B. Factoring (GCF, trinomials, difference of squares and cubes, sum of cubes, grouping) C Rationalizing (numerator and denominator) *D. Simplifying rational expressions *E. Solving algebraic equations and inequalities (linear, quadratic, higher order using synthetic division, rational, radical, and absolute value equations) F. Simultaneous equations II. Graphing and Functions *A. Lines (intercepts, slopes, write equations using point-slope and slope intercept, parallel, perpendicular, distance and midpoint formulas) B. Conic Sections (circle, parabola, ellipse, and hyperbola) *C Functions (definition, notation, domain, range, inverse, composition) *D. Basic shapes and transformations of the following functions (absolute value, rational, root, higher order curves, log, In, exponential, trigonometric. piece-wise, inverse functions) E. Tests for symmetry: odd, even III. Geometry A. Pythagorean Theorem B. Area Formulas (Circle, polygons, surface area of solids) C Volume formulas D. Similar Triangles * IV. Logarithmic and Exponential Functions *A. Simplify Expressions (Use laws of logarithms and exponents) *B. Solve exponential and logarithmic equations (include In as well as log) *C Sketch graphs *D. Inverses * V. Trigonometry **A. Unit Circle (definition of functions, angles in radians and degrees) B. Use of Pythagorean Identities and formulas to simplify expressions and prove identities *C Solve equations *D. Inverse Trigonometric functions E. Right triangle trigonometry *F. Graphs VI. Limits A. Concept of a limit B. Find limits as x approaches a number and as x approaches 00 * A solid working foundation in these areas is very important.

8 Calculus Prerequisite Problems Work the following problems on your own paper. Show all necessary work. I. Algebra A. Exponents: 1) (8x 3 yz)x (2X)3 4x X (yz%f B. Factor Completely: Hint: Factor as difference ofsquares first, then as 2) 9x 2 + 3x - 3xy - y (use grouping) the sum and difference ofcubes second. r 6) X -1-3x -2+ 2x -3 Hint: Factor out GCF x-3 first C. Rationalize denominator/numerator: 5/ 3/. ''I.' 5) 15.\-7:.' - 2.\/:: - 24x' 2 Hint: Factor GCF x 1/2 first 3-x jx+i + 1 7) 8) 1-,.Jx-2 D. Simplify the rational expression: (x + 1)3 (x - 2) + 3(x + 1 )2 9) (x + 1)4 E. Solve algebraic equations and inequalities Use synthetic division to help factor thefollowing, stateallfactors and roots. 10) p(x) =x3 + 4x 2 + x ) p(x) = 6x3-17x 2-16x + 7 X 12) Explain why "2 3 cannot be a root of f(x) =4x 5 + cx 3 - dx + 5, where c and d are integers. (hint: You can look at the possible rational roots.) 13) Explain why f(x) =x 4 + 7x 2 + x - 5 must have a root in the interval [0,1], (0 ~ x ~ 1) Check the graph and use signs of f(o) and f(l) to justify you're answer. Solve: You may use your graphingcalculator to check solutions. 14) (x + 3)2> 4 15) x+5 -- x-3 ~ 0 16) 3x 3-14x 2-5x ~ 0 (Factor first) 1 xl ) x < 18) ~ 0 19) > 0 x x+l x-l x-6 20) x 2 < 4 21) I 2x + 1 I <- 1 4

9 F. Solve the system. Solve the system algebraically and then check the solution by graphing each function and using your calculator to find the points of intersection. 22) x - Y + 1 = 0 23) x 2-4x + 3 = Y y-x 2 = -5 - x 2 + 6x - 9 = Y II. Graphing and Functions: A. Linear graphs: Write the equation of the line described below. 24) Passes through the point (2, -1) and has slope -3" 1. 25) Passes through the point (4, - 3) and is perpendicular to 3x + 2y = ) Passes through ( -1, - 2) and is parallel to y = 5" x -1. B. Conic Sections: Write the equation in standard[orm and identify the conic. 27) x = 4y 2 + 8y ) 4x 2-16x + 3y y + 52 = 0 C. Functions: Find the domain and range ofthefollowing. Note: domain restrictions - denominator i= 0, argument of a log or In > 0, radicand of even index must be ~ 0 range restrictions- reasoning, if all else fails, use graphing calculator 3 29) y=- 30) Y = log(x - 3) x-2 F+1 32) Y =~2x ) Y = I x - 5 I 34) domain only: y -- - x ) Given f(x) below, graph over the domain [-3, 3], what is the range? X ifx~o f (x) == 1 if -1 ~ x < 0 { x - 2 if x <-1

10 Find thecomposition/inverses as indicated below. Let f(x) = x 2 + 3x - 2 g(x) = 4x - 3 h(x) = lnx w(x)=~ 36) g -1(x) 37) h -l(x) 38) w -l(x), for x ~ 4 39) f(g(x)) 40) h(g(f(l))) 41) Does y = 3x 2-9 have an inverse function? Explain your answer. Let f(x) = 2x, g(x) = -x, and h(x) = 4, find 42) (f 0 g)(x) 43) (f 0 go h)(x) 44) Let s (x) =~ and t(x) = x 2, find the domain and range of (s 0 t )(x). D. Basic Shapes of Curves: Sketch the graphs. You may use your graphing calculator to verify your graph, but you should be able to graph the following by knowledge of the shape of the curve, by plotting a few points, and by your knowledge of transformations. 1 45) Y =~ 46) Y = lnx 47) Y = ~ 48) Y = I x - 2 I 1 x IT 49) Y = - 50) Y= 2 51) Y = 2- x 52) Y = 3 sin 2 (x - 6" ) x-2 x ) f(x) =../25 - x 2 if x < 0 x 2-25 if x ~ 0, x ~ 5 x-5 o if x =5 E. Even, Odd, Tests for Symmetry: Identify as odd, even, or neitherandjustify you're answer. To justify your answer you must show substitution using -x! It is not enough to simply check a number. Even: f (x) = f (-x) Odd: f (-x) = - f (x) 3 54) f(x) x 3 + 3x 55) lex) = x L 6x ) f(x) = x ~ x 2 X 57) f(x) = sin 2x 58) f(x) = x 2 + x 59) f(x) = x(x 2-1) ~ 60) f(x) = 2 x 61) What type of function (even or odd) results from the product of two even functions? odd functions?

11 Test for symmetry. Show substitution with variables to justify you're answer. Symmetric to y axis: replace x with - x and relation remains the same. Symmetric to x axis: replace y with - y and relation remains the same. Origin symmetry: replace x with - x, Y with - yard the relation is equivalent. 62) Y= x 4+ X 2 63) Y = sin(x) 64) Y = cos(x) ~ 65) x = y ) Y = x2 + 1 IV LOGARITHMIC AND EXPONENTIAL FUNCTIONS A. Simplify Expressions: 3 1 ( 1 ) 67) log, (11) 68) 3log33 - -log log ) log, ) lne 73) In 1 74) In e 2 B. Solve equations: 76) log x 2 - log 100 = log 1 77) 3 x+l = 15 V TRIGONOMETRY A. Unit Circle: Know the unit circle - radian and degree measure. Be prepared for a quiz. 78) State the domain, range and fundamental period for each function? a) y=sinx b) y= cosx c) y = tan x B. Identities: Simplify: 79) (tan 2x)(csC2 X). 1 (cscx)(tan-x)(sinx) 80) 1 cos- x 81) sec-x - tan- x 82) Verify: (1 - sin 2x)(1 + tan-x) = 1 C. Solve the Equations 83) cos-r = cos x + 2, 84) 2 sin(2x) = -{3, os x s 2rr 85) cos-x + sinx + 1 = 0, O:c:; x :c:; 2;r

12 D. Inverse Trig Functions: Note: Sin -1 x = Arcsin x 86) Arcsinl 88) ArccOs( ~J 89) sin[ ArCCOs( ~ J; 90. State domain and range for: Arcsin(x), Arccos (x), Arctan (x) E. Right Triangle Trig: Find the value of x. (Note: Degree measure!) x 10 A 180 ft B 60 ft c 93) The roller coaster car shown in the diagram above takes 23.5 sec. to go up the 23 degree incline segment AH and only 2.8 seconds to go down the drop from H to e. The car covers horizontal distances of 180 feet on the incline and 60 feet on the drop. Decimals in answer may vary. a. How high is the roller coaster above point B? b. Find the distances AH and He. c. How fast (in ftlsec) does the car go up the incline? d. What is the approximate average speed of the car as it goes down the drop? e. Assume the car travels along He. Is your approximate answer too big or too small? ( Advanced Mathematics, Richard G. Brown, Houghton Mifflin,1994, pg 336) F. Graphs: Identify the amplitude, period, horizontal, and vertical shifts of these functions. 94) Y = - 2sin(2x) 95) Y =-Jrcos( ~x + Jr)

13 G. Be able to do the following on your graphing calculator: Be familiar with the ICALCI commands; value, root, minimum, maximum, intersect. You may need to zoom in on areas of your graph to find the information. Answers should be accurate to 3 decimal places. Sketch graph Given the following function f(x) = 2x 4 - llx 3 - x x. 96. Find all roots. Note: Window x min: -10 x max: 10 scale 1 y min: -100 y max: 60 scale Find all local maxima. A local maximum or local minimum is a point on 98. Find all local minima. the graph where there is a highest or lowest point within an interval such as the vertex ofa parabola. 99. Find the following values: f (-1), f (2), f (0), f (.125) 100. Graph the following two functions and find their points of intersection using the intersect command on your calculator. y = x 3 + 5x 2-7x + 2 and y =.2x Window: x min:-lo x max: 10 scale 1 y min: -10 y max: 50 scale 0 VI. Functions and Models 101. The graphs of f and g are given. y -.- (a) State the values of f(-4) and g(3). (b) For what values of x if f(x)=g(x)? (c) Estimate the solution of the equation (d) On what interval is f decreasing? (e) State the domain and range of f. (f) State the domain and range of g. r-, i gj f/ - ~-~V ",V 2-1/1""-, /' _...'1 I -- ; '/, 0 i 2 \ / I -.!! I...L f(x) = The number N (in thousands) of cell ular phone subscribers in M alaysia is shown in the table. (Midyear estimates are given.) t N I 2461 I (a) Use the data to sketch a rough graph of N as a function of t. (b) Use your graph to estimate the number of cell-phone subscribers in Malaysia at midyear in 1994 and 1996.

14 103. If f(x) = 3x 2 -x+2, find f(2), f(-2), f(a), f(-a), f(a+1), 2f(a), f(a 2), [f(a)] 2, and f(a+h) Find the domain of each function. x a) j(x)=- 3x-l b) g(u)=..[;;+.j4-u 105. Find an expression for the bottom half of the parabola x+(y-1) 2 = A rectangle has perimeter 20 m. Express the area of the rectangle as a function of the length of one of its sides Find an expression for the function whose graph is the given curve e..,; -4 -.::: 108. Biologists have noticed that the chirping rate of crickets of a certain species is related to temperature, and the relationship appears to be very nearly linear. A cricket produces 113 chirps per minute at 70 F and 173 chirps per minute at 80 F. (a) Find a linear equation that models the temperature T as a function of the number of chirps per minute N. (b) What is the slope of the graph? What does it represent? (c) If the crickets are chirping at 150 chirps per minute, estimate the temperature At the surface of the ocean, the water pressure is the same as the air pressure above the water, 15lb/in2. Below the surface, the water pressure increases by 4.34Ib/in 2 for every 10 ft of descent. (a) Express the water pressure as a function of the depth below the ocean surface. (b) At what depth is the pressure 100 Ib/rn 2?

15 110. Classify each function as a power function, root function, polynomial (state its degree), rational function, algebraic function, or logarithmic function. (a) f(x) =if; (b) g(x)=jl-x 2 (d)r(x)=x x 3 +x Match each equation with its graph. Explain your choices. (Don't use a computer or graphing calculator) (a) y=x (b) y=x (c) y=x Suppose the graph of f is given. Write equations for the graphs that are obtained from the graph of f as follows. (a) Shift 3 units upward. (b) Shift 3 units downward. (c) Shift 3 units to the right. (d) Shift 3 units to the left. (e) Reflect about the x-axis. (f) Reflect about the y-axis, (g) Stretch vertically by a factor of 3. (h) Shrink vertically by a factor of The graph of y = f(x) is given. Match each equation with its graph and give reasons for your choices. (a) y = f(x-4) (b) Y = f(x)+3 (c) Y= 1/3 f(x) (d) Y = -f(x+4) (e) y = 2f(x+6)

16 114. The graph of f is given. Use it to graph the following functions. (a) y = f(2x) (b)=f( 1/2 x) (c) Y= f(-x) (d) Y= -f(-x) Graph the following, not by plotting points, but by starting with the graph of one of the standard functions and applying the appropriate transformations. 1. ( 1[1 y =3"sm X-tO 116. Find f+g, f - g, fg, and f/g and state their domains. l(x)=x 3+2x2, g(x)=3x Find the functions log, go I, 1 01, and gog and their domains. I(x) =sin X, g(x) =l--r; 118. Express the function in the form log. F(x) = (x 2 + l)'" 119. Use a graphing calculator to determine which of the given viewing rectangles produces the most appropriate graph of the function I(x) =10+ 25x-x 3 (a) [-4,4} by [-4,4] (b) [-10,10] by [-10,10] (c) [-20,20] by [-100,100] (d) [-100, 100] by [-200,200]

17 120. Use the given graphs of f and g to evaluate each expression, or explain why it is undefined. (a) f(g(2)) (b) g(f(o)) (c)(1 0 g)(o) (d) (go/)(6) (e) (g 0 g)(-2) (I) (/ 0 / )(4) and 122, Determine an appropriate viewing rectangle for the given function and use it to draw each graph I(x) =5 + 20x _x l(x)=~81-x Graph the ellipse 4x 2 + 2y 2 =1 by graphing the functions whose graphs are the upper and lower halves of the ellipse Find all solutions of the equation correct to three decimal places. x 3-9x 2-4 = Graph the given functions on a common screen. How are these graphs related? y=2 x, y=e x, y=5", y=20 x 126. Starting with the graph of y = e", write the equation of the graph that results from (a) shifting 2 units downward (b) shifting 2 units to the right (c) reflecting about the x-axis (d) reflecting about the y-axis (e) reflecting about the x-axis and then about the y-axis

18 127. Find the exponential function f(x) = Ca x whose graph is given Under ideal conditions a certain bacteria population is known to double every three hours. Suppose that there are initially 100 bacteria. (a) What is the size of the population after 15 hours? (b) What is the size of the population after t hours. (c) Estimate the size of the population after 20 hours. (d) Graph the population function and estimate the time for the population to reach 50, Determine whether this function is one-to-one. t@ ~.5 I ~.o I ~.6 I :.3 1-~ ~ Determine whether this function is one-to-one. -J Here is a verbal description of a function. Determine whether this function is oneto-one. P(t) is the height of a football t seconds after kickoff. For # , find a formula for the inverse of the function f(x)=~./lo-3x 133. I(x) = e' ) 134. Y = In(x+3) For # , find the exact value of each expression (non-calculator).

19 (a) log, 64 (b) log, (a) log., log., 80 (b) logs 10+ logs 20-3log s Express the given quantity as a single logarithm. 2ln4-ln The graph shown gives a salesman's distance from his home as a function of time on a certain day. Describe in words what the graph indicates about his travels on this day. Distance from home (miles) 8 A.M. to NOON P.M. Time (hours)

20 Answers: (Remember - you must show all of your work!) 1. 4xll/ 3y4/ 3z 2. (3x + 1)(3x - y) 3. (2x -1)(4x2 + 2x + 1)(2x + 1)(4x 2-2x +1) 4. 7(3x 2 + 4)(2x L 1) 5. x 1 / 2(3x - 4)(5x + 6) x x x 3(x - 2)(x - 1) 7.1+~ 9.. ~-1 (x + 1) (x -l)(x + 2)(x + 3); 1, -2, (x + 1)(2x - 7)(3x -1); -1, 2" ' not a possible rational root 13. f(o) = neg and f(l) = positive 14. x> -lor x < ~ x < x ~ -- or 0 ~ x ~ < x < 1 or x < [-3, -1) U [3, (0) 19. x> 6 or 1 < x < < x < ' 3 < x < -8' 22. (3,4), (-2, -1) 23. (2,-1),(3,0) y=--x y=-x Y =-x x = 4(y + 1)2-7 parabola (X_2)2 (y+4) =1 ellipse 29. 0: x 1= 2 R: Y 1= D: x > 3 Riall reals : all reals R: x ~ : x '?. 3/ 2 R: Y '? : all reals R: y ~ x >-1 and x ::j: R: - 5 s Y < - 3 or 0 ~ y s /-1(x) = X h -l(x) = ex Y= x x ~ 0, 39. f(g(x)) = 16x 2-12x In no, explain x D: -2 < x ~ 2 R: 0 < y <2 You must show work on these! 54. odd 55. even 56. odd 57. odd 58. neither 59. odd 60. even 61. even, even 62. symm y axis 63. origin 64. yaxis 65. x axis 66. yaxis / / log x = 10, log a) 0: all reals, R: -1 ~ x :::1, 2 JI b) D: allreals, R:-1:::x.:s,1,2JI c) D: x1=rr/2 R: allreals,rr sin-x yes 83. Jr 84. Jr Jr 7Jr 6'3'6' 4Jr Jr n 2

21 87. Jr 88. Jr 89. ~ 90. Arcsin(x) D: [-1,1] Range: [_ Jr, Jr] Arccos (x) D:[-l,l]. R: [O,II] Arctan (x) D: allreals R: [-~,;] sin hint: draw altitude, x IJ a. HB ft. b. AH ft., HC c. speed up approx ft./sec d. speed down approx ft./ sec e. Too small, since the distance traveled is longer the distance per foot would be greater. (Note: find average speed for the entire trip! ft /sec) 94. A: 2, P: II 95 A = Jr, Per. 4, V.s. = 0, h. s. = -2 Y= -Jrcos[~ (x + 2)] , 0, 2,5 97. rel max. (1.07,20.1) 98. rel. min (-.89, ), (3.94, - 88) 99. f( -1) = points of intersection - one is (-5.77,16.66) 100. Do not use trace to find points. Use CALC commands (a) -2,4 (b) -2 and 2 (c) x= -3 and 4 (d) [0,4] (e) d: [-4,4] r:[-2,3] (f) d: [-4,3] r:[0.5,4] 102. (b) 540 In 1994 and 1450 In f(2)=12, f(-2) = 16, f(a) = 2al\2 -a +2, f(-a)=3ai\2+a+2, f(a+1)=3ai\2+5a+4, 2f(a)=6aI\2-2a+4, f(2a)=12ai\2-2a+2, f(ai\2)=3ai\4 -a1\2 +2, [f(a))a2=9ai\4-6ai\3+13ai\2-4a+4, f(a+h)=3ai\2 +6ah+3hI\2 -a-h (a) (-00,1/3) U (1/3,00) (b) [0,4] 105. f(x) =1-h (domain: x:s; 0) X + 1 if -1 < x < A(L) = 10L -L1\2 domain 5<L< f(x) = 3 { --x+6 if 2<x~ (a) T=1/6 N + 307/6 (b) 1/6 means thatfor each increase of 6 cricket chirps per minute corresponds to an increase of 1 degree F. (c) 76 degrees F 109. (a) P=0.434d + 15 (b) approx. 196 feet 110. (a) root fct. (b) algebraic because it Is the root of a polynomial (c) polynomial of degree 9 (d) rational function 111. (a) matches with h (b) matches with f and (c) matches with g (a) y = f(x)+3 (b) Y= f(x) -3 (c) Y = f(x-3) (d) Y= f(x+3) (e) y = -f(x) (f) Y= f(-x) (g) Y= 3f(x) (h) Y= 1/3 f(x) 113. (a) graph 3 (b) graph 1 (c)graph 4 (d) graph 5 (e) graph (a) shrink horizontally by a factor of 2 (b) stretch horizontally by a factor of 2 (c) reflect the graph of f about the y-axis (d) reflect the graph of f about the y-axis, then about the x-axis (f+g)(x) =xi\3+5x'\2-1 d: all real numbers (f-g)(x)= xi\3-xi\2+1 d: all reals

22 (fg)(x)=3x A5+6xA4-xA3-2s A2 d: all reals (fjg)(x)=(x A3 +2x A2)j(3xA2-1) d: x cannot Equal ± ~ 117. (fog)(x)=sin(1-e) d:[o,oo) (joj)(x)=sin(sinx) d:(-oo,oo) (goj)(x)=i-.}sinx d:[o,tr],[2tr,3tr]efc (gog)(x)=i-ji-e d:(-oo,oo) 118. g(x)=x A2+1and F(x)=x AI c 120.(a) 5 (b) 3 (c)o (d) not defined (e) 4 (f) [-10,30] by [-50,150] 122. [-4,4] by [-1,4] 123. graph y = +sqr((1-4x A2)j2) and -sqr((1-4x A2)j2) 124. about All of these graphs approach 0 as x approaches negative infinity, all of them pass Through the point (0,1) and all of them are increasing and approach infinity as x Approaches infinity (a) y = eax -2 (b) Y= e A(x-2) (c) y = -eax (d) y = ea(-x) (e) y =- ea(-x) 127. f(x) = e(2 Ax) 128. (a) 3200 (b) Y= 100(2 A(tj3)) (c) 10,159 (d) 26.9 hrs No, does not pass horizontal line test No 131. A football will attain every height up to its maximum height twice: once on the way up, and again on the way down rr:: Hence,notltol.132. j-i(x)=_-x 2 +- d:[o,oo) 133. j-i(x)={tlnx 134. j-'(x)=e x (a) 6 (b) (a) 2 (b) In The salesman travels away from home from 8 am To 9 am and is then stationary until 10:00. The salesman travels farther away from 10 until noon. There is no change in his distance from home until 1:00, at which time the distance from home decreases until 3:00. Then the distance starts increasing again, reaching the maximum distance away from home at 5:00. There is no chance from 5 until 6, and then the distance decreases rapidly until 7:00 pm, at which time the salesman reaches home.